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Do you think you are normal?

Do you think you are normal?. Yes No I’m not average, but I’m probably within 2 standard deviations. Chapter 6. The Standard Deviation as a Ruler and the Normal Model. Normal Probability Plots.

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Do you think you are normal?

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  1. Do you think you are normal? • Yes • No • I’m not average, but I’m probably within 2 standard deviations.

  2. Chapter 6 The Standard Deviation as a Ruler and the Normal Model

  3. Normal Probability Plots • When you actually have your own data, you must check to see whether a Normal model is reasonable. • Looking at a histogram of the data is a good way to check that the underlying distribution is roughly unimodal and symmetric.

  4. Normal Probability Plots (cont.) • A more specialized graphical display that can help you decide whether a Normal model is appropriate is the Normal probability plot. • If the distribution of the data is roughly Normal, the Normal probability plot approximates a diagonal straight line. Deviations from a straight line indicate that the distribution is not Normal.

  5. Normal Probability Plots (cont.) • Nearly Normal data have a histogram and a Normal probability plot that look somewhat like this example:

  6. Normal Probability Plots (cont.) • A skewed distribution might have a histogram and Normal probability plot like this:

  7. The 68-95-99.7 Rule (cont.) • The following shows what the 68-95-99.7 Rule tells us:

  8. Three types of questions • What’s the probability of getting X or greater? • What’s the probability of getting X or less? • What’s the probability of X falling within in the range Y1 and Y2?

  9. IQ – Categorizes • Over 140 - Genius or near genius • 120 - 140 - Very superior intelligence • 110 - 119 - Superior intelligence • 90 - 109 - Normal or average intelligence • 80 - 89 - Dullness • 70 - 79 - Borderline deficiency • Under 70 - Definite feeble-mindedness

  10. Asking Questions of a Dataset • What is the probability that someone has an IQ over 100? • What is the probability that someone has an IQ lower than 85? • What is the probability that someone has an IQ between 85 and 130?

  11. Problem 8 • A • B • C

  12. About what percent of people should have IQ scores above 145? • .3% • .15% • 3% • 1.5% • 5% • 2.5%

  13. What percent of people should have iq scores below 130? • 95% • 5% • 2.5% • 97.5%

  14. Finding Normal Percentiles by Hand • When a data value doesn’t fall exactly 1, 2, or 3 standard deviations from the mean, we can look it up in a table of Normal percentiles. • Table Z in Appendix E provides us with normal percentiles, but many calculators and statistics computer packages provide these as well.

  15. Finding Normal Percentiles • Use the table in Appendix E • Excel • =NORMDIST(z-stat, mean, stdev, 1) • Online • http://davidmlane.com/hyperstat/z_table.html

  16. Finding Normal Percentiles by Hand (cont.) • Table Z is the standard Normal table. We have to convert our data to z-scores before using the table. • Figure 6.7 shows us how to find the area to the left when we have a z-score of 1.80:

  17. Categories of Retardation • Severity of mental retardation can be broken into 4 levels: • 50-70 - Mild mental retardation • 35-50 - Moderate mental retardation • 20-35 - Severe mental retardation • IQ < 20 - Profound mental retardation

  18. What percent of the population has an IQ of 20 or less? • 0.0001% • 0.0000% • 0.0004% • 0.04%

  19. What percent of the population has an IQ of 50 or less? • 0.0001% • 0.0000% • 0.0004% • 0.04%

  20. IQ - Categories • 115-124 - Above average (e.g., university students) • 125-134 - Gifted (e.g., post-graduate students) • 135-144 - Highly gifted (e.g., intellectuals) • 145-154 - Genius (e.g., professors) • 155-164 - Genius (e.g., Nobel Prize winners) • 165-179 - High genius • 180-200 - Highest genius • >200 - "Unmeasurable genius"

  21. What percent of the population has an IQ of 155 or more? • 99.99% • .01% • .9999 • .0001

  22. What percent of the population has an IQ of 120 or more? • 1.333 • .9082 • .0918 • 90.82% • 9.18%

  23. Average Height in Inches

  24. What fraction of men are less than 5’9 foot tall? • 50% • .1027 • 54.09% • 45.91%

  25. What fraction of women are less than 5’9 foot tall? • 1.78 • 96.25% • 3.75% • 45.91%

  26. From Percentiles to Scores: z in Reverse • Sometimes we start with areas and need to find the corresponding z-score or even the original data value. • Example: What z-score represents the first quartile in a Normal model?

  27. Height Problem • At what height does a quarter of men fall below? • At what height does a quarter of women fall below?

  28. From Percentiles to Scores: z in Reverse (cont.) • Look in Table Z for an area of 0.2500. • The exact area is not there, but 0.2514 is pretty close. • This figure is associated with z = -0.67, so the first quartile is 0.67 standard deviations below the mean.

  29. Z Score calculators • Excel • =NORMINV(prob, mean, stdev) • =NORMINV(0.25, 0, 1) • Online • Calculator • TI – 83/84 • TI-89

  30. TI- 83/84

  31. TI - 89

  32. Recovering the Mean and Standard Dev. • 17.5% 18 and under • 7.6% 65 and over • What is the mean and the standard deviation of the population?

  33. For next week… • Monday HW3 • Data Project Step 2 – Due Tuesday. • Thursday Quiz 2, covers HW2, HW3 and HW4 of the material learned in class.

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